Confused about setting up FQDN on Ubuntu 16.04 for outgoing mail server (exim)

I’m trying to follow this tutorial on setting up exim but for Ubuntu 16.04. I’m stuck on setting up the FQDN.

If I run hostname -f the result is mytime4caregivers. My DNS for this server is set up to host the site at https://mytime4caregivers.com. Does that mean my FQDN should be mytime4cargivers.mytime4caregivers.com?

Here’s my setup and what I’m trying to do:

• I’m hosting a Wordpress website
• I’m using Mailgun for incoming messages
• I only want to set up outgoing mail on the server so that I can receive messages from the server and so that Wordpress can do things like send out password reset emails, etc.
• I’m using Ubuntu 16.04, nginx, and PHP7

I think I should be updating /etc/hostname and /etc/hosts with the correct information, but I’m not sure what that should be. None of the tutorials that I’ve found seem to offer clear guidance. Thanks!

P.S. Here is my zone file per the DO DNS setup:

$ORIGIN mytime4caregivers.com.
$TTL 1800
mytime4caregivers.com. IN SOA ns1.digitalocean.com. hostmaster.mytime4caregivers.com. 1474757364 10800 3600 604800 1800
mytime4caregivers.com. 1800 IN NS ns1.digitalocean.com.
mytime4caregivers.com. 1800 IN NS ns2.digitalocean.com.
mytime4caregivers.com. 1800 IN NS ns3.digitalocean.com.
mytime4caregivers.com. 1800 IN A 104.131.68.19
email.mytime4caregivers.com. 1800 IN CNAME mailgun.org.
mytime4caregivers.com. 1800 IN MX 10 mxa.mailgun.org.
mytime4caregivers.com. 1800 IN MX 10 mxb.mailgun.org.
mytime4caregivers.com. 1800 IN TXT v=spf1 include:mailgun.org ~all
mailo._domainkey.mytime4caregivers.com. 1800 IN TXT "k=rsa; p=..."
www.mytime4caregivers.com. 1800 IN A 104.131.68.19